Semi-ring of half-closed-half-open intervals

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Definition

Let $a:= ({ a_i })_{ i = 1 }^{ n }\subseteq \mathbb{R}$ and $b:= ({ b_i })_{ i = 1 }^{ n }\subseteq \mathbb{R}$ be two finite sequences of the same length (namely $n\in\mathbb{N}$), we define $\[a,b\)$, a half-open-half-closed rectangle in $\mathbb{R}^n$ as follows:

• $\[a,b\):=[a_1,b_1)\times\cdots\times[a_n,b_n)\subset\mathbb{R}^n$ where $[\alpha,\beta):=\{x\in\mathbb{R}\ \vert\ \alpha\le x < \beta\}$

We denote the collection of all such half-open-half-closed rectangles by $\mathscr{J}^n$, $\mathscr{J}(\mathbb{R}^n)$ or, provided the context makes the dimensions obvious, simply just $\mathscr{J}$. Formally:

• $\mathscr{J}^n:=\{\[a,b\)\ \vert\ a,b\in\mathbb{R}^n\}$

Furthermore, we claim $\mathscr{J}^n$ is a [[semi-ring of sets][. For a proof of this claim see "Proof of claims" below.

Purpose

Probably the most important use case for this semi-ring is as the domain of a certain kind of pre-measure, namely a pre-measure on a semi-ring that serves as a precursor to the Lebesgue measure, which for the reader's curiosity we include the definition for here:

• $\lambda^n:\mathscr{J}^n\rightarrow\overline{\mathbb{R}_{\ge 0} }$ with $\lambda^n:\[a,b\)\mapsto\prod_{i=1}^n(b_i-a_i)$

In the case of $\mathscr{J}^1$ the Lebesgue measure is just the length of an interval, that is: $\lambda^1:[a,b)\mapsto (b-a)$ and for $\mathscr{J}^2$ it is the area of a rectangle, for $\mathscr{J}^3$ volume of a cuboid, and so forth.

We can then use the theorem: a pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring to get a normal pre-measure. Doing this is far easier than trying to define a pre-measure on the ring of sets generated by $\mathscr{J}^n$.

Once we have a pre-measure we can follow the usual path of extending pre-measures to measures

Proof of claims

[Expand]

Recall the definition of a semi-ring of sets

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$

A collection of sets, $\mathcal{F}$ is called a semi-ring of sets if:

1. $\emptyset\in\mathcal{F}$
2. $\forall S,T\in\mathcal{F}[S\cap T\in\mathcal{F}]$
3. $\forall S,T\in\mathcal{F}\exists(S_i)_{i=1}^m\subseteq\mathcal{F}$$\text{ pairwise disjoint}$$[S-T=\bigudot_{i=1}^m S_i]$ - this doesn't require $S-T\in\mathcal{F}$ note, it only requires that their be a finite collection of disjoint elements whose union is $S-T$.

In order to prove this we will first show that $\mathscr{J}^1$ (the collection of half-open-half-closed intervals of the form $[a,b)\subset\mathbb{R}$) is a semi-ring. Then we shall use induction on $n$ to show it for all $\mathscr{J}^n$

Grade: B

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It's a straightforward and fairly easy proof, but it isn't trivial. As such it is not marked as an easy proof, but nor does it have a high grade. See page 18 in Measures, Integrals and Martingales - René L. Schilling if stuck.